SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Critical parameter equations for degenerate parabolic equations coupled via nonlinear boundary flux

Si Xu* and Zifen Song

Author affiliations

Department of Mathematics, Jiangxi Vocational College of Finance and Economics, Jiujiang, Jiangxi, 332000, PR China

For all author emails, please log on.

Citation and License

Boundary Value Problems 2011, 2011:15  doi:10.1186/1687-2770-2011-15


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/15


Received:1 May 2011
Accepted:19 August 2011
Published:19 August 2011

© 2011 Xu and Song; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with the critical parameter equations for a degenerate parabolic system coupled via nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence parameter equation. The critical Fujita type is conjectured with the aid of some new results.

Mathematics Subject Classification (2000). 35K55; 35K57.

Keywords:
degenerate parabolic system; global existence; blow-up

1 Introduction

In this paper, we consider the following degenerate parabolic equations

u i t = ( u i p i ) x x , ( i = 1 , 2 , , k ) , x > 0 , 0 < t < T , (1.1)

coupled via nonlinear boundary flux

- ( u i p i ) x ( 0 , t ) = u i + 1 q i + 1 ( 0 , t ) , ( i = 1 , 2 , , k ) , u k + 1 : = u 1 , q k + 1 : = q 1 0 < t < T , (1.2)

with continuous, nonnegative initial data

u i ( x , 0 ) = u 0 i ( x ) , ( i = 1 , 2 , , k ) , x > 0 , (1.3)

compactly supported in ℝ+, where pi > 1, qi > 0, (i = 1, 2, ..., k) are parameters.

Parabolic systems like (1.1)-(1.3) appear in several branches of applied mathematics. They have been used to models, for example, chemical reactions, heat transfer, or population dynamics (see [1] and the references therein).

As we shall see, under certain conditions the solutions of this problem can become unbounded in a finite time. This phenomenon is known as blow-up, and has been observed for several scalar equations since the pioneering work of Fujita [2]. For further references, see the review by Leivine [3]. Blow-up may also happen for systems (see [4-7]). Our main interest here will be to determine under which conditions there are solutions of (1.1)-(1.3) that blow up and, in the blow-up case, the speed at which blowup takes place, and the localization of blow-up points in terms of the parameters pi, qi, (i = 1, 2, ..., k).

As a precedent, we have the work of Galaktionov and Levine [8], where they studied the single equation

u 1 t = ( u 1 p 1 ) x x , x > 0 , 0 < t < T , - ( u 1 p 1 ) x ( 0 , t ) = u 1 q 2 ( 0 , t ) , 0 < t < T , u 1 ( x , 0 ) = u 0 1 ( x ) , x > 0 . (1.4)

It was shown if 0 < q2 q0 = (p1 +1)/2, then all nonnegative solutions of (1.4) are global in time, while for q2 > q0 there are solutions with finite time blow-up. That is, q0 is the critical global existence exponent. Moreover, it was shown that qc := p1 + 1 is a critical exponent of Fujita type. Precisely, qc has the following properties: if q0 < q2 qc, the all nontrivial nonnegative solutions blow up in a finite time, while global nontrivial nonnegative solutions exist if q2 > qc.

We remark that there are some related works on the critical exponents for (1.1)-(1.3) in special cases.

In [9-11], the authors consider the case for pi = 1, (i = 1, 2, ..., k).

In [12], the authors consider the case for k = 2.

For the system (1.1)-(1.3), instead of critical exponents there are critical parameter equations, one for global existence and another of Fujita type. This is the content of our first theorem.

To state our results, we introduce some useful symbols. Denote by

A = 1 + p 1 - 2 q 2 0 0 0 0 0 0 1 + p 2 - 2 q 3 0 0 0 0 0 0 0 0 0 1 + p k - 1 - 2 q k - 2 q 1 0 0 0 0 0 1 + p k

A series of standard computations yield

det A = l = 1 k ( 1 + p l ) - l = 1 k 2 q l .

We shall see that det A = 0 is the critical global existence parameter equation. Let (α1, α2, ..., αk)T be the solution of the following linear algebraic system

A ( α 1 , α 2 , , α k - 1 , α k ) T = ( 1 , 1 , , 1 , 1 ) T ,

that is

α i = l = 1 k ( 1 + p l ) 2 q i [ l = 1 k ( 1 + p l ) - l = 1 k 2 q l ] m = i k + i - 1 j = i m 2 q j 1 + p j , q k + i = q i , p k + i = p i ( i = 1 , 2 , , k ) . (1.5)

We define

β i = 1 + ( p i - 1 ) α i 2 , ( i = 1 , 2 , , k ) . (1.6)

Theorem 1.1.

(I) If l = 1 k ( 1 + p l ) l = 1 k 2 q l (i.e. det A ≥ 0), every nonnegative solution of (1.1)-(1.3) is global in time.

(II) If l = 1 k ( 1 + p l ) < l = 1 k 2 q l (i.e. det A < 0) and there exists j (1 ≤ j k) such that αj + βj ≤ 0, then every nonnegative, nontrivial solution blows up in finite time.

(III) If l = 1 k ( 1 + p l ) < l = 1 k 2 q l (i.e. det A < 0), with αi + βi > 0 (i = 1, 2, ...,k), there exist nonnegative solutions with blow-up and nonnegative solutions that are global.

Therefore, the critical global existence parameter equation is

l = 1 k ( 1 + p l ) = l = 1 k 2 q l ( i . e . det A = 0 )

and the critical Fujita type parameter equation is

min { α 1 + β 1 , α 2 + β 2 , , α k + β k } = 0 .

The values of αi, βi (i = 1, 2, ..., k) are the exponents of self-similar solutions to problem (1.1)-(1.2). Such self-similar solutions are studied in Section 2, and play an important role in the proof of Theorem 1.1.

Let us observe that if we take k = 2, the critical parameter equations coincide with those found in [12].

The rest of this paper is organized as follows. In the next section, we study the existence of self-similar solutions of different type. In Section 3 we give some results concerning existence, comparison, monotonicity and uniqueness. In Section 4 we find the critical parameter equations (Theorem 1.1).

2 Self-similar solutions

In this section, we consider different kinds of self-similar solutions of problem (1.1)-(1.2). We have the following results.

Theorem 2.1. Let

u i ( x , t ) = ( T - t ) α i f i ( ξ i ) , ξ i = x ( T - t ) - β i , i = 1 , 2 , , k . (2.1)

If

l = 1 k ( 1 + p l ) < l = 1 k 2 q l , (2.2)

there is a self-similar solution of problem (1.1)-(1.2) blowing up in a finite time T > 0, of form (2.1). Moreover, the support of fi is + if βi > 0, and a compact set if βi ≤ 0 (i = 1, 2, ..., k).

Theorem 2.2. Let

u i ( x , t ) = t α i f i ( ξ i ) , ξ i = x t - β i , i = 1 , 2 , , k . (2.3)

(a) If

l = 1 k ( 1 + p l ) > l = 1 k 2 q l , (2.4)

then there exist functions fi positive in +, such that ui given in (2.3) is a self-similar solution of problem (1.1)-(1.2) global in time. These solutions have αi > 0 and thus their initial data are identically zero. Then βi < 0 (i = 1, 2, ...,k).

(b)If

α i + β i > 0 , i = 1 , 2 , , k ,

then there exist functions fi, compactly supported in +, such that ui given in (2.3) is a self-similar solution of problem (1.1)-(1.2) global in time. These solutions have αi < 0 and thus they decay to zero as t → ∞. Then βi > 0, and hence their supports expand as time increases.

Remark 2.2. If there exists j (1 ≤ j k) such that α j + βj ≤ 0, there are no profiles fi L1(ℝ+) such that ui (i = 1, 2, ..., k,) given by (2.3) is a solution. Indeed

0 u j ( x , t ) d x = t α j + β j 0 f j ( ξ j ) d ξ j .

Then, if αj + βj ≤ 0, the mass of uj would not increase, a contradiction.

Theorem 2.3. Let

u i ( x , t ) = e α i t f i ( ξ i ) , ξ i = x e - β i t , i = 1 , 2 , , k . (2.5)

If

l = 1 k ( 1 + p l ) = l = 1 k 2 q l , (2.6)

for any α1 > 0, there is a self-similar solution of problem (1.1)-(1.2) global in time of form (2.5) where

α i = α 1 j = 2 i 1 + p j - 1 2 q j ( i = 2 , , k ) , β i = ( p i - 1 ) α i 2 ( i = 1 , 2 , , k ) . (2.7)

Moreover, the supports of fi (i = 1, 2, ..., k) are compact.

Remark 2.3. The solutions are in principle weak. However, if they are positive everywhere, they are also classical.

In order to prove these theorems, we will use the following results of Gilding and Peletier (see [13-15]):

Theorem 2.4. Let a, b, V ∈ ℝ and U ≥ 0. For fixed a and b, let SA denote the set of values of (U, V) such that there exists a weak, nonnegative, compactly supported solution f1 of

( f 1 p 1 ) ( η ) + a η f 1 ( η ) = b f 1 ( η ) , 0 < η < , (2.8)

f 1 ( 0 ) = U , (2.9)

( f 1 p 1 ) ( 0 ) = V , (2.10)

and let S B denote the set of values (U, V) for which there exists a bounded, positive, classical solution f1 of (2.8)-(2.10).

(a) If b < 0 and 2a + b < 0, then S A = {(0, 0)} and SB = Ø.

(b) If b < 0 and 2a + b = 0, then S A = {(0, V): 0 ≤ V < ∞} and S B = Ø.

(c) If b ≤ 0 and 2a + b > 0, then there exists a unique V* such that S A = { ( U , U ( p 1 + 1 ) 2 V * )   :   0 U < 1 } and S B = {(U, V): 0 ≤ U < ∞, U ( p 1 + 1 ) 2 V * < V < } , where V* > 0 if a + b < 0, V* = 0 if a + b = 0, and V* < 0 if a + b > 0.

(d) If b > 0 and a ≥ 0, then there exists a unique V* < 0 such that S A = { ( U , U ( p 1 + 1 ) 2 V * )   :   0 U < 1 } and S B = Ø.

(e) If b > 0 and a < 0, or b = 0 and a ≤ 0, then S A = {(0, 0)} and there exists a unique V* such that S B = {(U, U(p1+1)/2V*): 0 ≤ U < ∞}, where V* < 0 if b > 0 and V* = 0 if b = 0.

Moreover, for each (U, V) ∈ S A S B there exists at most one weak solution of (2.8)-(2.10).

Remark 2.4. In the case where a = ((p1 - 1)/2)b > 0, we have V* = -1. This is a consequence of the existence for a self-similar solution of exponential form for the scalar problem (1.4) with q2 = (p1 + 1)/2 (see [8]).

Proof of Theorem 2.1. We consider solutions of form (2.1). Imposing that the porous equations (1.1) are fulfilled, we get the following relations for the parameters:

α i - 1 = α i p i - 2 β i , i = 1 , 2 , , k . (2.11)

On the other hand, the boundary conditions (1.2) imply that

α i p i - β i = α i + 1 q i + 1 , i = 1 , 2 , , k , α k + 1 = α 1 , q k + 1 = q 1 . (2.12)

Solving the linear systems (2.11)-(2.12), we get that αi, βi (i = 1, 2, ..., k) are given by (1.5) and (1.6). Therefore, αi < 0 (i = 1, 2, ..., k) if and only if l = 1 k ( 1 + p l ) < l = 1 k 2 q l . On the other hand, the profiles must satisfy

( f i p i ) ( ξ i ) - β i ξ i f i ( ξ i ) = - α i f i ( ξ i ) , i = 1 , 2 , , k , (2.13)

plus the boundary conditions

- ( f i p i ) ( 0 ) = f i + 1 q i + 1 ( 0 ) , i = 1 , 2 , , k , q k + 1 = q 1 , f k + 1 = f 1 . (2.14)

Then fi satisfy (2.8) with coefficients ai = -βI, bi = -αi (i = 1, 2, ..., k). Thus, Theorem 2.4 parts (d) and (e) says that there is an one-parameter family (parameter Ui) of (2.8) satisfying

f i ( 0 ) = U i , ( f i p i ) ( 0 ) = U i ( p i + 1 ) 2 V * i ,

where V*i < 0 (i = 1, 2, ..., k) are constants. The profile fi has compact support if βi ≤ 0 and is positive in ℝ+ if βi > 0. We choose Ui such that the boundary conditions (2.14) are fulfilled, that is

- U i ( p i + 1 ) 2 V * i = U i + 1 q i + 1 , i = 1 , 2 , , k , U k + 1 = U 1 , q k + 1 = q 1 .

Taking logarithms, this is equivalent to

A ( ln U 1 , ln U 2 , , ln U k - 1 , ln U k ) T = - 2 ( ln | V * 1 | , ln | V * 2 | , , ln | V * k - 1 | , ln | V * k | ) T . (2.15)

As l = 1 k ( 1 + p l ) l = 1 k 2 q l (i.e. det A ≠ 0), the above system has a unique solution. □

Proof of Theorem 2.2. We are considering solutions of the form (2.3). Imposing that the equations (1.1) and that boundary conditions (1.2) are fulfilled, we get that the exponents should satisfy the relations (2.11)-(2.12). Hence they are given by (1.5)-(1.6). Moreover, the boundary conditions for the profiles are given by (2.14). However, the equations for the profiles are now different:

( f i p i ) ( ξ i ) + β i ξ i f i ( ξ i ) = α i f i ( ξ i ) , i = 1 , 2 , , k . (2.16)

Thus, fi satisfy (2.8) with coefficients ai = βi, bi = αi (i = 1, 2, ..., k).

(I) If αi > 0, that is, if (2.4) holds, then βi < 0 (i = 1, 2, ..., k). Therefore, applying Theorem 2.4 part (d) as in the proof of Theorem 2.1, and taking the solutions of (2.15) as values for parameters, we obtain that there exist positive profiles fi (i = 1, 2, ..., k) solving (2.16) and satisfying (2.14).

(II) If αi < 0 and αi + βi > 0 (i = 1, 2, ..., k), we can apply Theorem 2.4 part (c) as in the proof of Theorem 2.1 and taking the solutions of (2.15) as the parameters, we obtain that there exist compactly supports profiles fi (i = 1, 2, ..., k) solving (2.16) and satisfying the boundary conditions (2.14).

Proof of Theorem 2.3. We are considering solutions of the form (2.5). Though the boundary conditions (1.2) impose (2.12) again, now equations (1.1) impose different relations for the exponents. Namely

α i = α i p i - 2 β i , i = 1 , 2 , , k . (2.17)

Thus,

A ( α 1 , α 2 , , α k - 1 , α k ) T = ( 0 , 0 , , 0 , 0 ) T . (2.18)

There are nontrivial solutions of (2.18) if and only if l = 1 k ( 1 + p l ) = l = 1 k 2 q l (i.e. det A = 0). In this case, β1, αI, βi (i = 2, ...,k) are related to α1 by (2.7).

The boundary conditions for the profiles are again given by (2.14), while the equations for the profiles are given by (2.16). If α1 > 0, then β1, αi, βi > 0 (i = 2, ..., k) and βi = ((pi - 1)/2)αi (i = 1, ..., k). Hence, using Remark 2.4, we have solutions of (2.16) with V*i = -1 (i = 1, 2, ...,k). Choosing one of the solutions of (2.15) with right-hand side zero (again we are using l = 1 k ( 1 + p l ) = l = 1 k 2 q l (i.e. det A = 0)), we obtain that there exist compactly supported profiles fi (i = 1, 2, ..., k) solving (2.16) and satisfying (2.14).

3 Existence and uniqueness

First, we state a theorem that guarantees the existence of a solution. It can be obtained using a standard monotonicity argument following ideas from [16].

Theorem 3.1. Given continuous, compactly supported initial data u0i(x) (i = 2, ..., k), there exists a local in time continuous weak solution of (1.1)-(1.3). Moreover, if the initial data are smooth and compatible in sense that

- ( u 0 i p i ) x ( 0 ) = u 0 i + 1 q i + 1 ( 0 ) , i = 2 , , k , u 0 k + 1 ( x ) = u 0 1 ( x ) ,

then the solution has continuous time derivatives down to t = 0.

Proof. Let us consider the Neumann problem

w t = ( w r ) x x , x > 0 , 0 < t < τ , - ( w r ) x ( 0 , t ) = h ( t ) , 0 < t < τ , w ( x , 0 ) = w 0 ( x ) , x > 0 , (3.1)

with r > 1. We define the operator M q i + 1 : C ( [ 0 , τ ] ) C ( [ 0 , τ ] ) as M q i + 1 ( h ) ( t ) = w q i + 1 ( 0 , t ) , where w(x, t) is the unique solution of (3.1) with r = pi and initial condition w0(x) = u0i(x) i = 1 , 2 , , k , M q k + 1 = M q 1 , w q k + 1 = w q 1 .

It has been proved in [17] that M q i ( i = 1 , 2 , . . . , k ) is continuous and compact. Moreover, they are order preserving.

Now let A ( h ) = M q k M q k - 1 M q 2 M q 1 ( h ) . Using the method of monotone iterations, one can prove that there exist τ > 0 such that A has a fixed point in C([0, τ]). This fixed point provides us with a continuous weak solution of (1.1)-(1.3) up to time τ.

In order to obtain the regularity of the solution with compatible initial data, we only have to observe that the solution of (3.1) is regular if - ( w 0 r ) x = h ( 0 ) (see [18]).

Remark 3.1. If the initial data are compactly support, the solution ui (i = 1, 2, ..., k) also has compact support as long as it exists.

Remark 3.2. If the initial data are nontrivial, we can assume that they satisfy u0i(x) > 0 (i = 1, 2, ..., k). If not, ui(0, t) (i = 1, 2, ..., k) eventually become positive (compare with a Barenblatt solution of the corresponding equation).

Next, we define what called a subsolution and a supersolution for (1.1)-(1.2).

Definition 3.1. ( u 1 , u 2 , , u k - 1 , u k ) is a subsolution of (1.1)-(1.2) if it satisfies

u i t ( u i p i ) x x x > 0 , 0 < t < T , i = 1 , 2 , , k , (3.2)

- ( u i p i ) x ( 0 , t ) u i + 1 q i + 1 ( 0 , t ) , u k + 1 q k + 1 = u 1 q 1 , 0 < t < T , i = 1 , 2 , , k . (3.3)

Definition 3.2. We call ( ū 1 , ū 2 , , ū k - 1 , ū k ) a supersolution of (1.1)-(1.2) of it satisfies (3.2)-(3.3) with the opposite inequalities.

With these definitions of super and subsolutions, we can state a comparison lemma.

Lemma 3.1 Let ( ū 1 , ū 2 , , ū k - 1 , ū k ) be a supersolution and ( u 1 , u 2 , , u k - 1 , u k ) be a subsolution. If

u i ( x , 0 ) ū i ( x , 0 ) , i = 1 , 2 , , k ,

with

u i ( 0 , 0 ) ū i ( 0 , 0 ) , i = 1 , 2 , , k ,

then

u i ( x , t ) ū i ( x , t ) , i = 1 , 2 , , k ,

as long as both super and subsolutions exist.

Proof. It is standard, therefore we omit the details. Assume that the result is false. Let t0 be the maximum time such that

u i ( x , t ) ū i ( x , t ) , i = 1 , 2 , , k ,

up to t0. This time t0 must be positive, by continuity. At that time, we must have u j ( 0 , t 0 ) = ū j ( 0 , t 0 ) for some j (1 ≤ j k). Let us assume that u 1 ( 0 , t 0 ) = ū 1 ( 0 , t 0 ) . Now the result follows by an application of Hopf's lemma. Indeed, ū 1 - u 1 satisfies a uniformly parabolic equation in a neighborhood of x = 0, attains a minimum at (0, t0), and the corresponding flux is greater or equal than zero, a contradiction.

Now we state a lemma that guarantees that, for certain initial data, the solution of (1.1)-(1.3) increases in time.

Lemma 3.2 Let u0i(x) be the initial data for (1.1) -(1.3) such that u0i(x) are smooth, satisfy the compatibility condition at the boundary and ( u 0 i p i ) x x 0 . Then ui(x, t) increases in time, i.e., uit(x, t) ≥ 0 (i = 1, 2, ...,k).

Proof. Let wi = uit. Then, as the solutions are smooth (Theorem 3.1), we can differentiate to obtain the (w1, ..., wk) is a solution of

w i t = ( p i u i p i - 1 w i ) x x , i = 1 , 2 , k , (3.4)

- ( p i u i p i - 1 w i ) x ( 0 , t ) = q i + 1 u i + 1 q i + 1 - 1 w i + 1 ( 0 , t ) , q k + 1 = q 1 , u k + 1 = u 1 , w k + 1 = w 1 , (3.5)

with initial data satisfying

w i ( x , 0 ) 0 , i = 1 , 2 , , k .

To conclude the proof we apply the maximum principle. Due to the degeneration of the equations this cannot be done directly. A standard regularization procedure is needed (see [8] for details).

Next, we deal with the problem of uniqueness versus non-uniqueness for (1.1)-(1.3) on the case of vanishing initial data (u0i(x) = 0, i = 1, 2, ..., k).

Theorem 3.2

(a) Let l = 1 k ( 1 + p l ) > l = 1 k 2 q l . Then there exists a nontrivial solution with zero initial data that becomes positive at × = 0 instantaneously. Then there is no uniqueness for problem (1.1)-(1.3) with zero initial data.

(b) Let l = 1 k ( 1 + p l ) l = 1 k 2 q l . Then the solution of (1.1)-(1.3) with zero initial data is unique.

Proof.

(a) The self-similar solutions constructed in Theorem 2.2 become positive at x = 0 instantaneously.

(b) We can construct small supersolution with the aid of the self-similar ones of exponential form that we found in Theorem 2.3. First, choose q ̃ 1 q 1 such that 2 q ̃ 1 l = 2 k 2 q l = l = 1 k ( 1 + p l ) .

ū i ( x , t ) = e α i ( t + τ ) f i ( x e - β i ( t + τ ) ) , i = 1 , 2 , , k ,

where α1 > 0 is arbitrary and β1, αi, βi, (i = 2, ..., k) are given by (2.7). Now we observe that ( ū 1 , ū 2 , , ū k - 1 , ū k ) be a supersolution is a supersolution of (1.1)-(1.3) as long as u ¯ 1 ( 0 , t ) 1 . By the comparison Lemma 3.1, we obtain that every solution has initial data identically zero satisfies

ū i ( x , t ) u i ( x , t ) , i = 1 , 2 , , k .

As ū i can be chosen as small as we want (using τ negative and large enough) we conclude that ū i 0 ( i = 1 , 2 , , k ) .

4 Blow-up versus global existence

We devote this section to prove Theorem 1.1. We borrow ideas from [8]. However, the fact that we are dealing with a system instead of a single equation forces us to develop a significantly different proof. We will organize the proof in several lemmas.

Our first lemma proves part (I) of Theorem 1.1.

Lemma 4.1. If l = 1 k ( 1 + p l ) l = 1 k 2 q l (i.e. det A ≥ 0), every nonnegative solution of (1.1)-(1.3) is global in time.

Proof. It is enough to construct global supersolutions with initial data as large as needed. We achieve this with the aid of the self-similar solutions of exponential form that we found in Theorem 2.3.

First we choose q ̃ 1 q 1 such that 2 q ̃ 1 l = 2 k 2 q l = l = 1 k ( 1 + p l ) and we let

ū i ( x , t ) = e α i ( t + τ ) f i ( x e - β i ( t + τ ) ) , i = 1 , 2 , , k ,

where α1 > 0 is arbitrary and β1, αi, βi, (i = 2, ..., k) are given by (2.7). Now we observe that ( ū 1 , ū 2 , , ū k - 1 , ū k ) is a supersolution of (1.1)-(1.3) as long as ū 1 ( 0 , t ) 1 . This can be done by choosing τ large enough. This also allows to assume ū i ( x , 0 ) u 0 i ( x ) ( i = 1 , 2 , , k ) . Then, by the comparison Lemma 3.1, we obtain that every solution is global.

Now we construct subsolutions with finite time blow-up.

Lemma 4.2. Let l = 1 k ( 1 + p l ) < l = 1 k 2 q l (i.e. det A < 0), then there exist compactly supported functions gi (i = 1, 2, ..., k), such that

u i ( x , t ) = ( T - t ) α i g i ( ξ i ) , ξ i = x ( T - t ) - β i , i = 1 , 2 , , k ,

is a subsolution of (1.1)-(1.2).

Proof. To satisfy (3.2) and (3.3), we need that

( g i p i ) ( ξ i ) - α i g i ( ξ i ) + β i ξ i g i ( ξ i ) , i = 1 , 2 , , k , - ( g i p i ) ( 0 ) g i + 1 q i + 1 ( 0 ) , i = 1 , 2 , , k , q k + 1 = q 1 , g k + 1 = g 1 .

We choose

g i ( ξ i ) = A i ( a i - ξ i ) + 1 ( p i - 1 ) , i = 1 , 2 , , k .

Inserting this in the equation, we get

p i ( p i - 1 ) 2 A i p i - 1 - α i ( a i - ξ i ) + - β i p i - 1 ξ i f o r 0 ξ i a i , i = 1 , 2 , , k .

Hence, it is enough to impose

p i ( p i - 1 ) 2 A i p i - 1 - α i a i + | β i | p i - 1 a i , i = 1 , 2 , , k ,

that is

C i A i p i - 1 a i , i = 1 , 2 , , k . (4.1)

The boundary conditions impose

p i p i - 1 A i p i a i 1 ( p i - 1 ) A i + 1 q i + 1 a i + 1 q i + 1 ( p i + 1 - 1 ) , i = 1 , 2 , , k , A k + 1 = A 1 , q k + 1 = q 1 , a k + 1 = a 1 . (4.2)

Let

b i = A i a i 1 ( p i - 1 ) , i = 1 , 2 , , k .

Then conditions (4.2) become

p i p i - 1 A i p i - 1 b i b i + 1 q i + 1 , i = 1 , 2 , , k , b k + 1 q k + 1 = b 1 q 1 . (4.3)

We fix bi = 1 (i = 1, 2, ⋯, k) and then Ai large enough (and thus ai small) to satisfy (4.1) and (4.3).

Corollary 4.1 Let l = 1 k ( 1 + p l ) < l = 1 k 2 q l (i.e. det A < 0). Then there exist solutions of (1.1)-(1.3) that blow up in a finite time.

Proof. We only have to apply Lemma 3.1, to obtain that every solution (u1, ⋯, uk) that begins above the subsolutions provided by Lemma 4.2 has finite time blow-up.

Lemma 4.3 Let l = 1 k ( 1 + p l ) < l = 1 k 2 q l (i.e. det A < 0). If there exists j (1 ≤ j k) such that αj + βj ≤ 0, then every nontrivial solution of (1.1)-(1.3) blows up in finite time.

Proof. Without loss of generality, we consider the case α1 + β1 ≤ 0.

Assume that there exists a global nonnegative nontrivial solution of (1.1)-(1.3), we make the following change of variables

φ i ( ξ i , τ ) = ( 1 + t ) - α i u i ( ξ i ( 1 + t ) β i , t ) , τ = log ( 1 + t ) , i = 1 , 2 , , k . (4.4)

These functions satisfy

φ i τ = ( φ i p i ) ξ i ξ i + β i ξ i φ i ξ i - α i φ i , i = 1 , 2 , , k , (4.5)

- ( φ i p i ) ξ i ( 0 , τ ) = φ i + 1 q i + 1 ( 0 , τ ) , i = 1 , 2 , , k , φ k + 1 q k + 1 = φ 1 q 1 . (4.6)

As ui(x, t) (i = 1, 2, ..., k) are by hypothesis global, the same is true for φi (i = 1, 2, ..., k,). We will construct a solution ( φ ^ 1 , , φ ^ k ) to system (4.5)-(4.6) increasing with time, with initial data ( φ ^ 0 1 , , φ ^ 0 k ) such that φ ^ 0 i ( ξ i ) u i ( ξ i , 0 ) ( i = 1 , 2 , , k ) . We will prove that ( φ ^ 1 , , φ ^ k ) cannot exists globally, thus contradicting the global existence of (u1, ⋯, uk). In order to achieve our goal, we use an adaptation for systems of the general monotonicity for single quasilinear equation described in [19].

We take initial data ( φ ^ 0 1 , , φ ^ 0 k ) satisfying

( φ ^ 0 i p i ) ξ i ξ i + β i ξ i ( φ ^ 0 i ) ξ i - α i φ ^ 0 i 0 , i = 1 , 2 , , k ,

and the compatibility conditions

- ( φ ^ 0 i p i ) ξ i ( 0 ) = φ ^ 0 i + 1 q i + 1 ( 0 ) , i = 1 , 2 , , k , φ ^ 0 k + 1 q k + 1 = φ ^ 0 1 q 1 .

Hence, arguing as in Lemma 3.2, we have that φ ^ i τ 0 ( i = 1 , 2 , , k ) .

Following an idea for scalar equation from [8], we set

φ ^ 0 1 ( ξ 1 ) = h ( ξ 1 + b ) ,

where h is the Barenblatt profile

h ( ξ 1 ) = a p 1 ( c - ξ 1 2 ) + 1 ( p 1 - 1 ) .

Then we have

( φ ^ 0 1 p 1 ) ξ 1 ξ 1 + β 1 ξ 1 ( φ ^ 0 1 ) ξ 1 - α 1 φ ^ 0 1 = - 1 p 1 + 1 b h ξ 1 ( ξ 1 + b ) + ( β 1 - 1 p 1 + 1 ) ξ 1 h ξ 1 ( ξ 1 + b ) + ( - α 1 - 1 p 1 + 1 ) h ( ξ 1 + b ) .

The last expression is nonnegative if β1 - 1/(p1 + 1) ≤ 0 and -α1 - 1/(p1 + 1) ≥ 0. But these two conditions are equivalent α1 + β1 ≤ 0.

Now we take α ̃ i , β ̃ i > 0 such that α ̃ i α i , β ̃ i β i ( i = 1 , 2 , , k ) . We take as φ ^ 0 i a solution to

( φ ^ 0 i p i ) = - β ̃ i ξ i φ ^ 0 i + α ̃ i φ ^ 0 i , i = 1 , 2 , , k .

There is one-parameter family of solution to this equation (see Theorem 2.4), with φ ^ 0 i 0 , φ ^ 0 i 0 ( i = 2 , , k ) . Hence,

( φ ^ 0 i p i ) - β i ξ i φ ^ 0 i + α i φ ^ 0 i , i = 2 , , k .

Moreover,

φ ^ 0 i ( 0 ) = U i , ( φ ^ 0 i p i ) ( 0 ) = U i ( p i + 1 ) 2 V * i , i = 2 , , k ,

where V*i < 0 is a constant and Ui is the free parameter.

We still have to control the boundary conditions. In order to do this, we choose the constants c, b and Ui (i = 2, ...,k) conveniently. They have to satisfy

2 p 1 a p 1 p 1 - 1 b ( c - b 2 ) 1 ( p 1 - 1 ) = U 2 q 2 , b ( 0 , c 1 2 ) , - V * i U i ( p i + 1 ) 2 = a p 1 q i + 1 ( c - b 2 ) q i + 1 ( p 1 - 1 ) , i = 2 , , k .

Thus, we choose

U 2 = c 2 b 2 q 3 ( 2 q 2 q 3 - p 2 - 1 ) , U i = c i b 2 ( p 2 - 1 ) q i + 1 ( ( p i + 1 ) ( 2 q 2 q 3 - p 2 - 1 ) ) , ( i = 3 , , k , q k + 1 = q 1 ) , c = b 2 + γ b ( p 1 - 1 ) ( p 2 - 1 ) ( 2 q 2 q 3 - p 2 - 1 ) ,

where ci (i = 2, ..., k) and γ are positive constants. Taking b small enough, the initial data ( φ ^ 0 1 , , φ ^ 0 k ) is below (u1(ξ1,0), ...,uk(ξk, 0)). This can be done as u0i (i = 1, 2, ... k) can be assumed to be positive at the origin.

To conclude the proof, we will show that ( φ ^ 1 , , φ ^ k ) converge to a self-similar profile that does not exist in this range of parameters.

Lemma 4.4. There exists j (1 ≤ j k) such that

lim τ φ ^ j ( ξ j , τ ) = φ ̃ j ( ξ j ) < , ξ j > 0 . (4.7)

Proof. It is clear that φ ^ i ξ i 0 ( i = 1 , 2 , , k ) . Let us suppose that

φ ^ i ( ξ i , τ ) u n i f o r m l y i n ( 0 , ξ i 0 ) , i = 1 , 2 , , k .

In the original variables ( u ^ 1 , , u ^ k ) , we have that for any M > 0 there is a value such that

( 1 + t 0 ) α i M u ^ i ( x , t 0 ) f o r 0 < x ( 1 + t 0 ) - β i < ξ i 0 , i = 1 , 2 , , k . (4.8)

Now we will check that, under these conditions, we can put one of the blowing up subsolutions constructed in Lemma 4.2 below these data. This would lead to a contradiction, as ( u ^ 1 , , u ^ k ) is global. In order to do this, we need

( 1 + t 0 ) a 1 M A 1 a 1 1 ( p 1 - 1 ) T α 1 , ξ 1 0 ( 1 + t 0 ) β 1 a 1 T β 1 . (4.9)

The first equation says that the height at x = 0 of u ^ 1 is bigger than that of u 1 , and the second says that the support of u ^ 1 is bigger than the support of u 1 . Imposing analogous conditions for u ^ i and u i ( i = 2 , , k ) we get

( 1 + t 0 ) a i M A i a i 1 ( p i - 1 ) T α i , ξ i 0 ( 1 + t 0 ) β i a i T β i . (4.10)

Taking T = 1 + t0, then ai small enough and Ai large enough (i = 1, 2, ..., k), and then M large, then the 2k conditions (4.9)-(4.10) are fulfilled.

Let us remark this parametric evolution comparison method to prove global non-existence for arbitrary data first introduced in [20], for scalar quasilinear heat equation.

End of the proof of Lemma 4.3. Let us assume that (4.7) holds. Using standard arguments, see [8], we may pass to the limit to obtain that

( φ ̃ 1 p 1 ) ξ 1 ξ 1 + β 1 ξ 1 φ ̃ 1 ξ 1 - α 1 φ ̃ 1 = 0 . (4.11)

Let z = φ ̃ 1 p 1 , then

z ξ 1 ξ 1 + β 1 p 1 ξ 1 z ( 1 - p 1 ) p 1 z ξ 1 0 .

Hence, in (0, ξ10), z c > 0,

z ξ 1 ξ 1 C z ξ 1 .

We conclude that z and therefore φ ̃ 1 cannot be unbounded at ξ1 = 0. In particular, 0 < φ ̃ 1 ( 0 ) C . Then, considering the regularity of φ ̃ 1 in the region where φ ̃ 1 > 0 , we can pass to the limit in the boundary condition for ( φ ^ 1 p 1 ) ξ 1 to obtain that

- ( φ ̃ 1 p 1 ) ξ 1 ( 0 ) = φ ̃ 2 q 2 ( 0 ) . (4.12)

However, as α1 + β1 ≤ 0, problem (4.11)-(4.12) does not have a nontrivial solution, see Theorem 2.4.

If (4.7) holds for some j > 1, we can proceed as before to obtain that φ ̃ j ( 0 ) < . Thus, we can pass to the limit in the boundary condition for φ ^ j , obtaining

- ( φ ̃ j p j ) ξ j ( 0 ) = φ ̃ j + 1 q j + 1 ( 0 ) .

As φ ̃ j + 1 ( 0 ) φ ̃ j + 1 ( ξ j + 1 ) , this implies that φ ̃ j + 1 is finite for every ξj+1 ≥ 0. We get the same contradiction as before.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

We would like to thank Professor Dimitru Motreanu, Christopher Rualizo and the referees for their valuable comments and suggestions.

References

  1. Pao, CV: Nonlinear Parabolic and Elliptic Equations. Plenum, New York (1992)

  2. Fujita, H: On the blowing up of solutions for the Cauchy problem for ut = Δu + u1+α. J Fac Sci Univ Tokyo Sec IA Math. 16, 105–113 (1996)

  3. Levine, HA: The role of critical exponents in blow up theorems. SIAM Rev. 32, 262–288 (1990). Publisher Full Text OpenURL

  4. Andreucci, D, Herrero, MA, Velázquez, JJL: Liouville theorems and blow-up behaviour in a semilinear reaction diffusion systems. Ann Inst H Poincaré Anal Nonlinéaire. 14, 1–53 (1997)

  5. Escobedo, M, Levine, HA: Critical blow-up and global existence numbers for a weakly coupled system of reaction-diffusion equations. Arch Rational Mech Anal. 129, 47–100 (1995). Publisher Full Text OpenURL

  6. Rossi, JD, Wolanski, N: Blow-up vs. global existence for a semilinear reaction-diffusion system in a bounded domain. Comm Partial Differential Equations. 20, 1991–2004 (1995). Publisher Full Text OpenURL

  7. Wang, S, Xie, CH, Wang, MX: Note on critical exponents for a system of equations coupled in the boundary conditions. J Math Anal Appl. 218, 313–324 (1998). Publisher Full Text OpenURL

  8. Galaktionov, VA, Levine, HA: On critical Fujita exponents for heat equations with nonlinear flux boundary conditions on the boundary. Israel J Math. 94, 1250–146 (1996)

  9. Lin, ZG: Blowup behaviors for diffusion system coupled though nonlinear boundary conditions in a half space. Sci China Ser A Math. 47, 72–82 (2004)

  10. Pedersen, M, Lin, ZG: Blow-up analysis for a system of heat equations coupled through a nonlinear boundary condition. Appl Math Lett. 14, 171–176 (2001). Publisher Full Text OpenURL

  11. Wang, MX: The blow-up rates for systems of heat equations with nonlinear boundary conditions. Sci China Ser A Math. 46, 169–175 (2003). Publisher Full Text OpenURL

  12. Quirós, F, Rossi, JD: Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions. Indiana Univ Math J. 50, 629–654 (2001)

  13. Gilding, BH: On a class of similarity solutions of the porous media equation III. J Math Anal Appl. 77, 381–402 (1980). Publisher Full Text OpenURL

  14. Gilding, BH, Peletier, LA: On a class of similarity solutions of the porous media equation. J Math Anal Appl. 55, 351–364 (1976). Publisher Full Text OpenURL

  15. Gilding, BH, Peletier, LA: On a class of similarity solutions of the porous medium equation II. J Math Anal Appl. 57, 522–538 (1977). Publisher Full Text OpenURL

  16. Bénilan, Ph, Cortázar, C, Elgueta, M: Uniqueness and non uniqueness of the solutions of a mixed boundary value problem for the porous medium equation. Rev Un Mat Argentina. 37, 10–15 (1991)

  17. Cortázar, C, Elgueta, M, Vázquez, JL: Diffusivity determination in non linear diffusion. Eur J Appl Math. 2, 159–169 (1991). Publisher Full Text OpenURL

  18. Kalashnikov, AS: Some problems of the qualitative theory of nonlinear degenerate parabolic equations of second order. Uspekhi Mat Nauk. 42, 135–176 (Russian); Russian Math. Surv. 42, 169-222 (1987) (English transl.) (1987)

  19. Galaktionov, VA: Blow-up for quasilinear heat equations with critical Fujita's exponents. Proc R Soc Edinburg Sect A. 124, 517–525 (1994)

  20. Galaktionov, VA, Kurdyumov, SP, mikhailov, AP, Samarskii, AA: Unbounded solutions of the cauchy problem for the parabolic equation ut = ▽(uσu) + uβ. Dokl Akad Nauk SSSR Ser Math Phys. 252, 1362–1364 (Russian); Soviet Phys. Dokl. 25, 458-459 (1980) (English transl.) (1980)